Spain's World Cup-winning captain, 30, is in the final year of his City contract with no indication he is ready to sign an extension after seven years at the Etihad.
Real had appeared best placed to sign him, having offered around £43m for his services, but no agreement has been reached and Barcelona are now understood to have stepped up their interest, putting them in pole position ahead of José Mourinho's Real Madrid.
Barcelona's need in midfield has been sharpened by a medial knee ligament tear to Frenkie de Jong that is expected to keep him out for a lengthy spell, while a switch to the Camp Nou would also hold personal appeal for Rodri given his frosty relationship with Real dating back to his Ballon d'Or win over Vinicius Jr in autumn 2024.
City, now managed by Maresca following Pep Guardiola's departure this summer, have had to weigh their own contract offer carefully given Rodri has missed much of the past two seasons through injury. Real, meanwhile, have already committed heavily elsewhere this window, agreeing a fee in the region of £115.8m with RB Leipzig for 19-year-old winger Yan Diomande.
England midfielder Elliot Anderson has already arrived at the Etihad for £116m, and City could move for a further midfield reinforcement should Rodri push through a departure before the end of the window.
The situation marks a reversal from earlier reporting, which had long cast Real Madrid as Rodri's clear preference if he were to leave City.
Since joining from Atletico Madrid in 2019, Rodri has established himself as one of City's most important players, winning four Premier League titles, the Champions League and two FA Cups, while his World Cup-winning form for Spain this year has only sharpened suitors' interest despite the back surgery and knee problems that disrupted his past two seasons.
Real's resources have also been strained by an active summer window, having already added Marc Cucurella, Bernardo Silva and Ibrahima Konaté before their move for Diomande, while Barcelona are also working to bring in a replacement for Robert Lewandowski, leaving both Spanish giants to weigh how much they are willing to commit to a deal that could still cost in the region of £75m were City to be persuaded to sell.